Brouwer Fixed Point Theory

Abstract

A topological space Y has the fixed point property, abbreviated fpp, if every map (continuous function) f: Y → Y has a fixed point, that is, f(y) = y for some y ∈ Y. The fixed point property is a topological property in the sense that it is preserved by homeomorphisms. That is, it’s easy to see that if a space Y has the fpp and Z is homeomorphic to Y, then Z also has the fpp. The fixed point theorem quoted in Chapter 1 as the key to the topological proof of the Cauchy-Peano theorem states that a compact, convex subset of a normed linear space has the fpp. We’ll prove that theorem and more in Chapter 4. The proof is accomplished in two steps: first prove a finite-dimensional fixed point theorem, then generalize to normed linear spaces. This chapter will be devoted to the first of these steps.